Solve Inequality: Algebraic Proof of a<b<c<1/12

[Lizu]

hi! i need help for this inequality
1. $a\in\mathbb{N}*~and~ \frac{a}{a+1}<\frac{a+1}{a+2}<\frac{a+2}{a+3}$
show that : $\frac{1}{2}*\frac{4}{5}*...*\frac{2005}{2006}*\frac{2008}{2009}<\frac{1}{12}$
Here i have stoped. Please tell me if is corect what i have done so far and how to continue , or another idea to solve

 

[pasmith]

Use your inequality to obtain an upper bound for $P_1^3$.

 

[BvU]

Hello Lizu,

Good start. Except the last factors: P₁ last one is ${2008\over 2009}$ etc.
Make it so that your P₁ is the product you are after
You can still show $\ \ P_1P_2 P_3 = {1\over 2011}\ \$ and $\ \ P_1<P_2<P_3 \ \$.

So if you can show $P_1^3 < P_1P_2P_3$ you are in business !

[edit] ah, PA was faster. Nice exercise !

 

[Lizu]

Thank you !